Tensor Computations of Stochastic Dynamic Fields

This article attempts to delve a kind of quantities—tensor for a stochastic dynamic domain, making its principles of computations such as gradient, divergence and interaction. The notation "stochastic dynamic field (SDF)", proposed by the author, refers to such a dynamic field that it appears or annihilates, moves, acts or interacts at random. The study aims to consider not only randomness of vector fields in a continuous dynamic system, but also discontinuity with time variable t, i.e. more randomness for t, in a discontinuous dynamic system. To do so, representation forms of tensors areproperly transformed, which apply multiple-points tensor, and introduce the quantity of compound tensor like an expression of quantum state of combination of uncertainties. The work bears out some new results: (1) discontinuous mutation, even emerge or disappear, of a point in a tensor field, analyzed using a tensor with matrix structure, would not defer tensor computations of that field, while wouldn't be computed commonly using accustomed representation of tensor with only a single-point structure. (2) Random quantity relations between phrases, no matter smooth nearby, non-connected or not, all can be computed. (3) Some classical physical laws analyzing field properties of space-relevant factors such as Euler equation and Gauss theorem are available in new forms with a random nature. (paper)